Helper functions for plotting the geometric representation of matroids¶

AUTHORS:

• Jayant Apte (2014-06-06): initial version

Note

This file provides functions that are called by show() and plot() methods of abstract matroids class. The basic idea is to first decide the placement of points in $mathbb{R}^2$ and then draw lines in geometric representation through these points. Point placement procedures such as addtripts, addnontripts together produce (x,y) tuples corresponding to ground set of the matroid in a dictionary. These methods provide simple but rigid point placement algorithm. Alternatively, one can build the point placement dictionary manually or via an optimization that gives aesthetically pleasing point placement (in some sense. This is not yet implemented). One can then use createline function to produce sequence of 100 points on a smooth curve containing the points in the specified line which inturn uses scipy.interpolate.splprep and scipy.interpolate.splev. Then one can use sage’s graphics primitives line, point, text and points to produce graphics object containg points (ground set elements) and lines (for a rank 3 matroid, these are flats of rank 2 of size greater than equal to 3) of the geometric representation of the matroid. Loops and parallel elements are added as per conventions in [Oxl2011] using function addlp. The priority order for point placement methods used inside plot() and show() is as follows:

1. User Specified points dictionary and lineorders

2. cached point placement dictionary and line orders (a list of ordered lists) in M._cached_info (a dictionary)

3. Internal point placement and orders deciding heuristics If a custom point placement and/or line orders is desired, then user can simply specify the custom points dictionary as:

M.cached info = {'plot_positions':<dictionary_of _points>,
'plot_lineorders':<list of lists>}


REFERENCES¶

• [Oxl2011] James Oxley, “Matroid Theory, Second Edition”. Oxford University Press, 2011.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0], [0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
sage: pos_dict= {0: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.666666666666667),
....: 7: (3,3), 8: (4,0), 9: (-1,1), 10: (-2,-2)}
sage: M1._cached_info={'plot_positions': pos_dict, 'plot_lineorders': None}
sage: matroids_plot_helpers.geomrep(M1, sp=True)
Graphics object consisting of 22 graphics primitives

sage.matroids.matroids_plot_helpers.addlp(M, M1, L, P, ptsdict, G=None, limits=None)

Return a graphics object containing loops (in inset) and parallel elements of matroid.

INPUT:

• M – A matroid.
• M1 – A simple matroid corresponding to M.
• L – List of elements in M.groundset() that are loops of matroid M.
• P – List of elements in M.groundset() not in M.simplify.groundset() or L.
• ptsdict – A dictionary containing elements in M.groundset() not necessarily containing elements of L.
• G – (optional) A sage graphics object to which loops and parallel elements of matroid $$M$$ added .
• limits– (optional) Current axes limits [xmin,xmax,ymin,ymax].

OUTPUT:

A 2-tuple containing:

1. A sage graphics object containing loops and parallel elements of matroid M
2. axes limits array

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: M=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0],[0, 0, 1, 1, 1, 0, 1,0,0]])
sage: [M1,L,P]=matroids_plot_helpers.slp(M)
sage: G.show(axes=False)


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.addnontripts(tripts_labels, nontripts_labels, ptsdict)

Return modified ptsdict with additional keys and values corresponding to nontripts.

INPUT:

• tripts – A list of 3 ground set elements that are to be placed on vertices of the triangle.
• ptsdict – A dictionary (at least) containing ground set elements in tripts as keys and their (x,y) position as values.
• nontripts– A list of ground set elements whose corresponding points are to be placed inside the triangle.

OUTPUT:

A dictionary containing ground set elements in tripts as keys and their (x,y) position as values allong with all keys and respective values in ptsdict.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: ptsdict={'a':(0,0),'b':(1,2),'c':(2,0)}
....:         ['d','e','f'],ptsdict)
sage: setprint(ptsdict_1)
{'a': [0, 0], 'b': [1, 2], 'c': [0, 2], 'd': [0.6666666666666666, 1.0],
'e': [0.6666666666666666, 0.8888888888888888],
'f': [0.8888888888888888, 1.3333333333333333]}
....:         ['d','e','f','g','h'],ptsdict)
sage: setprint(ptsdict_2)
{'a': [0, 0], 'b': [1, 2], 'c': [0, 2], 'd': [0.6666666666666666, 1.0],
'e': [0.6666666666666666, 0.8888888888888888],
'f': [0.8888888888888888, 1.3333333333333333],
'g': [0.2222222222222222, 1.0],
'h': [0.5185185185185185, 0.5555555555555555]}


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.createline(ptsdict, ll, lineorders2=None)

Return ordered lists of co-ordinates of points to be traversed to draw a 2D line.

INPUT:

• ptsdict – A dictionary containing keys and their (x,y) position as values.
• ll – A list of keys in ptsdict through which a line is to be drawn.
• lineorders2– (optional) A list of ordered lists of keys in ptsdict such that if ll is setwise same as any of these then points corresponding to values of the keys will be traversed in that order thus overriding internal order deciding heuristic.

OUTPUT:

A tuple containing 4 elements in this order:

1. Ordered list of x-coordinates of values of keys in ll specified in ptsdict.
2. Ordered list of y-coordinates of values of keys in ll specified in ptsdict.
3. Ordered list of interpolated x-coordinates of points through which a line can be drawn.
4. Ordered list of interpolated y-coordinates of points through which a line can be drawn.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: ptsdict={'a':(1,3),'b':(2,1),'c':(4,5),'d':(5,2)}
sage: x,y,x_i,y_i=matroids_plot_helpers.createline(ptsdict,
....: ['a','b','c','d'])
sage: [len(x), len(y), len(x_i), len(y_i)]
[4, 4, 100, 100]
sage: G = line(zip(x_i, y_i),color='black',thickness=3,zorder=1)
sage: G+=points(zip(x, y), color='black', size=300,zorder=2)
sage: G.show()
sage: x,y,x_i,y_i=matroids_plot_helpers.createline(ptsdict,
....: ['a','b','c','d'],lineorders2=[['b','a','c','d'],
....: ['p','q','r','s']])
sage: [len(x), len(y), len(x_i), len(y_i)]
[4, 4, 100, 100]
sage: G = line(zip(x_i, y_i),color='black',thickness=3,zorder=1)
sage: G+=points(zip(x, y), color='black', size=300,zorder=2)
sage: G.show()


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.geomrep(M1, B1=None, lineorders1=None, pd=None, sp=False)

Return a sage graphics object containing geometric representation of matroid M1.

INPUT:

• M1 – A matroid.
• B1 – (optional) A list of elements in M1.groundset() that correspond to a basis of M1 and will be placed as vertices of the triangle in the geometric representation of M1.
• lineorders1 – (optional) A list of ordered lists of elements of M1.grondset() such that if a line in geometric representation is setwise same as any of these then points contained will be traversed in that order thus overriding internal order deciding heuristic.
• pd - (optional) A dictionary mapping ground set elements to their (x,y) positions.
• sp – (optional) If True, a positioning dictionary and line orders will be placed in M._cached_info.

OUTPUT:

A sage graphics object of type <class ‘sage.plot.graphics.Graphics’> that corresponds to the geometric representation of the matroid.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M)
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M,lineorders1=[['f','e','d']])
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.it(M, B1, nB1, lps)

Return points on and off the triangle and lines to be drawn for a rank 3 matroid.

INPUT:

• M – A matroid.
• B1– A list of groundset elements of M that corresponds to a basis of matroid M.
• nB1– A list of elements in the ground set of M that corresponds to M.simplify.groundset() \ B1.
• lps– A list of elements in the ground set of matroid M that are loops.

OUTPUT:

A tuple containing 4 elements in this order:

1. A dictionary containing 2-tuple (x,y) co-ordinates with M.simplify.groundset() elements that can be placed on the sides of the triangle as keys.
2. A list of 3 lists of elements of M.simplify.groundset() that can be placed on the 3 sides of the triangle.
3. A list of elements of $$M.simplify.groundset()$$ that cane be placed inside the triangle in the geometric representation.
4. A list of lists of elements of M.simplify.groundset() that correspond to lines in the geometric representation other than the sides of the triangle.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers as mph
sage: M=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0],
....: [0, 1, 0, 1, 0, 1, 1,0],[0, 0, 1, 1, 1, 0, 1,0]])
sage: N=M.simplify()
sage: B1=list(N.basis())
sage: nB1=list(set(M.simplify().groundset())-set(B1))
sage: pts,trilines,nontripts,curvedlines=mph.it(M,
....: B1,nB1,M.loops())
sage: print(pts)
{1: (1.0, 0.0), 2: (1.5, 1.0), 3: (0.5, 1.0), 4: (0, 0), 5: (1, 2),
6: (2, 0)}
sage: print(trilines)
[[3, 4, 5], [2, 5, 6], [1, 4, 6]]
sage: print(nontripts)
[0]
sage: print(curvedlines)
[[0, 1, 5], [0, 2, 4], [0, 3, 6], [1, 2, 3], [1, 4, 6], [2, 5, 6],
[3, 4, 5]]


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.line_hasorder(l, lodrs=None)

Determine if an order is specified for a line

INPUT:

• l – A line specified as a list of ground set elements.
• lordrs – (optional) A list of lists each specifying an order on a subset of ground set elements that may or may not correspond to a line in geometric representation.

OUTPUT:

A tuple containing 2 elements in this order:

1. A boolean indicating whether there is any list in lordrs that is setwise equal to l.
2. A list specifying an order on set(l) if 1. is True, otherwise an empty list.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: matroids_plot_helpers.line_hasorder(['a','b','c','d'],
....: [['a','c','d','b'],['p','q','r']])
(True, ['a', 'c', 'd', 'b'])
sage: matroids_plot_helpers.line_hasorder(['a','b','c','d'],
....: [['p','q','r'],['l','m','n','o']])
(False, [])


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.lineorders_union(lineorders1, lineorders2)

Return a list of ordered lists of ground set elements that corresponds to union of two sets of ordered lists of ground set elements in a sense.

INPUT:

• lineorders1 – A list of ordered lists specifying orders on subsets of ground set.
• lineorders2 – A list of ordered lists specifying orders subsets of ground set.

OUTPUT:

A list of ordered lists of ground set elements that are (setwise) in only one of lineorders1 or lineorders2 along with the ones in lineorder2 that are setwise equal to any list in lineorders1.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: matroids_plot_helpers.lineorders_union([['a','b','c'],
....: ['p','q','r'],['i','j','k','l']],[['r','p','q']])
[['a', 'b', 'c'], ['p', 'q', 'r'], ['i', 'j', 'k', 'l']]

sage.matroids.matroids_plot_helpers.posdict_is_sane(M1, pos_dict)

Return a boolean establishing sanity of posdict wrt matroid M.

INPUT:

• M1 – A matroid.
• posdict – A dictionary mapping ground set elements to (x,y) positions.

OUTPUT:

A boolean that is True if posdict indeed has all the required elements to plot the geometric elements, otherwise False.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
sage: pos_dict= {0: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
sage: matroids_plot_helpers.posdict_is_sane(M1,pos_dict)
True
sage: pos_dict= {1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
....: 4: (0.5, 1.0), 5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
sage: matroids_plot_helpers.posdict_is_sane(M1,pos_dict)
False


Note

This method does NOT do any checks. M1 is assumed to be a matroid and posdict is assumed to be a dictionary.

sage.matroids.matroids_plot_helpers.slp(M1, pos_dict=None, B=None)

Return simple matroid, loops and parallel elements of given matroid.

INPUT:

• M1 – A matroid.
• pos_dict – (optional) A dictionary containing non loopy elements of M as keys and their (x,y) positions. as keys. While simplifying the matroid, all except one element in a parallel class that is also specified in pos_dict will be retained.
• B – (optional) A basis of M1 that has been chosen for placement on vertices of triangle.

OUTPUT:

A tuple containing 3 elements in this order:

1. Simple matroid corresponding to M1.
2. Loops of matroid M1.
3. Elements that are in $$M1.groundset()$$ but not in ground set of 1 or in 2

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
sage: [M,L,P]=matroids_plot_helpers.slp(M1)
sage: M.is_simple()
True
sage: setprint([L,P])
[{7}, {8, 9, 10}]
sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
sage: posdict= {8: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
sage: [M,L,P]=matroids_plot_helpers.slp(M1,pos_dict=posdict)
sage: M.is_simple()
True
sage: setprint([L,P])
[{7}, {0, 9, 10}]


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.tracklims(lims, x_i=[], y_i=[])

Return modified limits list.

INPUT:

• lims – A list with 4 elements [xmin,xmax,ymin,ymax]
• x_i – New x values to track
• y_i – New y values to track

OUTPUT:

A list with 4 elements [xmin,xmax,ymin,ymax]

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: matroids_plot_helpers.tracklims([0,5,-1,7],[1,2,3,6,-1],
....: [-1,2,3,6])
[-1, 6, -1, 7]


Note

This method does NOT do any checks.

sage.matroids.matroids_plot_helpers.trigrid(tripts)

Return a grid of 4 points inside given 3 points as a list.

INPUT:

• tripts – A list of 3 lists of the form [x,y] where x and y are the Cartesian co-ordinates of a point.

OUTPUT:

A list of lists containing 4 points in following order:

1. Barycenter of 3 input points.
• 2,3,4. Barycenters of 1. with 3 different 2-subsets of input points respectively.

EXAMPLES:

sage: from sage.matroids import matroids_plot_helpers
sage: points=matroids_plot_helpers.trigrid([[2,1],[4,5],[5,2]])
sage: print(points)
[[3.6666666666666665, 2.6666666666666665],
[3.222222222222222, 2.888888888888889],
[4.222222222222222, 3.222222222222222],
[3.5555555555555554, 1.8888888888888886]]
`

Note

This method does NOT do any checks.